# Properties

 Conductor 8041 Order 560 Real No Primitive Yes Parity Even Orbit Label 8041.fs

# Related objects

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(8041)

sage: chi = H[41]

pari: [g,chi] = znchar(Mod(41,8041))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Conductor = 8041 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 560 Real = No sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = Yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = Even Orbit label = 8041.fs Orbit index = 149

## Galois orbit

sage: chi.sage_character().galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Values on generators

$$(6580,2366,562)$$ → $$(e\left(\frac{3}{10}\right),e\left(\frac{11}{16}\right),e\left(\frac{1}{7}\right))$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 12 $$1$$ $$1$$ $$e\left(\frac{219}{280}\right)$$ $$e\left(\frac{129}{560}\right)$$ $$e\left(\frac{79}{140}\right)$$ $$e\left(\frac{117}{560}\right)$$ $$e\left(\frac{1}{80}\right)$$ $$e\left(\frac{53}{80}\right)$$ $$e\left(\frac{97}{280}\right)$$ $$e\left(\frac{129}{280}\right)$$ $$e\left(\frac{111}{112}\right)$$ $$e\left(\frac{89}{112}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{560})$$